Vine copulas have a computational complexity that is increasing quadratically in the number of variables.This complexity can be reduced by focusing on the sub-class of truncated vine copulas,which use only a limited number of vine trees.We propose a new approach to select the adequate number of trees,such that the vine copula is still sufficiently flexible to provide a good fit to given data.The approach is based on fit indices,as used for structural equation models,to measure the goodness of a fitted truncated model.To select such truncated models,we propose methods to effectively explore the search space of truncated vine copulas,so that we are able to improve over simple greedy solutions,which were previously considered in the literature.This new selection approach is evaluated in a simulation study as well as in applications to data sets of financial returns.